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real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"fbd1042429dcc9c8b42d341e3a8238f56f75f6d134c0f6ad79b5a40822251435","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-688f70ea6f82c256","role":"sequence","name":"Stirling2 triangle with scaled diagonals (powers of 8).","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:            24,              2;\n  1:           448,             96,              3;\n  2:          7680,           3200,            240,              4;\n  3:        126976,          92160,          12480,            480,             5;\n  4:       2064384,        2465792,         537600,          35840,           840,            6;\n  5:      33292288,       63307776,       20901888,        2150400,         85120,         1344,          7;\n  6:     534773760,     1585971200,      763822080,      113885184,       6773760,       177408,       2016,        8;\n  7:    8573157376,    39132856320,    26821263360,     5573836800,     467496960,     18063360,     336000,     2880,      9;\n  8:  137304735744,   956334866432,   916979712000,   258715156480,   29407150080,   1572544512,   42577920,   591360,   3960,   10;\n  9: 2197949513728, 23226646265856, 30777853083648, 11571245875200, 1734937149440, 123351072768, 4559622144, 91238400, 982080, 5280, 11;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(128+16*x+x^2)*P(n-2,x) - (64*x+8*x^2)*d/dx P(n-2,x) + (24+2*x)*P(n-1,x) + 8*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 24 + 2*x, P(1,x) = 448 + 96*x + 3*x^2.","mathematica":"P[0, x_] := 24 + 2*x;\nP[1, x_] := 448 + 96*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[-(128 + 16*x + x^2)*P[n - 2, x] - (64*x + 8*x^2)*D[P[n - 2, x], x] + (24 + 2*x)*P[n - 1, x] + 8*x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 24 + 2*x; v[2] = 448 + 96*x + 3*x^2; for(n = 2, N, my(i = n + 1); v[i] = -(128 + 16*x + x^2)*v[i - 2] - (64*x + 8*x^2)*deriv(v[i - 2], x) + (24 + 2*x)*v[i - 1] + 8*x*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-688f70ea6f82c256","steps":["D"],"edges":["e-3949aa490ce01678"],"text":"A075503 \u2192 D"},"headline":"A075503 \u2192 D","chain":null}